A condition of cooperation. Games on network

Natural selection is often regarded as a result of severe competition. Defect seems beneficial for a single individual in many cases.However, cooperation is observed in many levels of biological systems ranging from single cells to animals, including human society. We have yet known that in unstructured populations, evolution favors defectors over cooperators. On the other hand, there have been much interest on evolutionary games^1,2^ on structured population and on graphs^3-16^. Structures of biological systems and societies of animals can be taken as networks. They discover that network structures determine results of the games. Together with the recent interest of complex networks^17,18^, many researchers investigate real network structures. Recently even economists study firms' transactions structure^19^. Seminal work^11^ derives the condition of favoring cooperation for evolutionary games on networks, that is, benefit divided by cost, _b/c_, exceeds average degree, (_k_). Although this condition has been believed so far^20^, we find the condition is _b/c_ (_k~nm~_) instead. _k~nm~_ is the mean nearest neighbor degree. Our condition enables us to compare how network structure enhances cooperation across different kinds of networks. Regular network favors most, scale free network least. On ideal scale free networks, cooperation is unfeasible. We could say that (_k_) is the degree of itself, while _k~nm~_ is that of others. One of the most interesting points in network theory is that results depend not only on itself but also on others. In evolutionary games on network, we find the same characteristic

Knowledge Spillover on Complex Networks

Most growth theories have focused on R&D activities. Although R&D significantly influences economic growth, the spillover effect also has a considerable influence. In this paper, we study knowledge spillover among agents by representing it as network structures. The objective of this study is to construct a framework to treat knowledge spillover as a network. We introduce a knowledge spillover equation, solve it analytically to find a workable solution. It has mainly three properties: (1) the growth rate is common for all the agents only if they are linked to the entire network regardless of degrees, (2) the TFP level is proportional to degree, and (3) the growth rate is determined by the underlying network structure. We compare growth rate among representative networks: regular, random, and scale-free networks, and find the growth rate is the greatest in scale-free network. We apply this framework, i.e., knowledge spill over equation, to the problem of firms forming a network endogenously and show how distance and region size affect the economic growth. We also apply the framework to network formation mechanism. The aim of our paper is not just showing results, but in constructing a framework to study spillover by network.

Primal-dual distance bounds of linear codes with application to cryptography

Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower bound and an upper bound on $N(d,d^\perp)$. Further, for small values of $d$ and $d^\perp$, we determine $N(d,d^\perp)$ and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.Comment: 6 pages, using IEEEtran.cls. To appear in IEEE Trans. Inform. Theory, Sept. 2006. Two authors were added in the revised versio

Network Structure of Japanese Firms Hierarchy and Degree Correlation: Analysis from 800,000 Firms

We found hierarchical structure and negative degree correlation in firms' transaction network. The network consists of 800,000 Japanese firms. We also summarize other features of the network and discuss why studying network structure is important. We also found scale free distribution in undirected network. --Network of firms,scale free network,complex network,hierarchy,degree correlation

Network structure of Japanese firms. Scale-free, hierarchy, and degree correlation: analysis from 800,000 firms

We analyze fundamental characteristics of the inter-firm transaction network through the data of 800,000 Japanese firms. We find that there exists a hierarchical structure and a negative degree correlation in this transaction network. We also find that this undirected network is a scale-free network. We bring to light these characteristics of the network and discuss why there is an important need to conduct research work on the actual network structure. --Network of firms,scale-free network,complex network,hierarchy,degree correlation